Exchange energy and generalized polarization in the presence of spin-orbit coupling in two dimensions

We discuss a general form of the exchange energy for a homogeneous system of interacting electrons in two spatial dimensions, which is particularly suited in the presence of a generic spin-orbit interaction. The theory is best formulated in terms of a generalized fractional electronic polarization. Remarkably, we find that a net generalized polarization does not necessarily translate into an increase in the magnitude of the exchange energy, a fact that, in turn, favors unpolarized states. Our results account qualitatively for the findings of recent experimental investigations.

Figure 1

Plot of the exchange energy per particle ${\mathcal{E}}_{xn}(r_s,p)$ (in Ry units) as a function of $p$. Here, $r_s=1$. The different values of $n$ are noted, and the limiting curve $n=\infty$ is also displayed (dashed line).

Figure 2

Plot of the fractional generalized polarization $p_{\mathit{min}}$ as function of the parameter $\overline{\gamma }$ for different values of $r_s$. Here, $n=2$. The increase of the depopulation value ${\overline{\gamma }}_d$ with $r_s$ is manifested.